{
  "abstract": "For a Hausdorff locally quasi-convex abelian group $G$, let $\\C(G)$ be the poset of all Hausdorff locally quasi-convex group topologies on its underlying group having the same continuous characters as $G$. We prove that, whenever $G$ is non-precompact, $\\C(G)$ contains an order-isomorphic copy of $(\\Pow(\\cont),\\subseteq)$, where $\\cont=2^{\\aleph_0}$. The embedding takes values between the Bohr topology and the original topology. Consequently, every infinite discrete abelian group $D$ satisfies $|\\C(D)|=\\width\\C(D)=2^{2^{|D|}}$. The discrete reduction for locally compact abelian groups then yields exact cardinality and width formulas for all such groups; in particular, both invariants equal $2^{\\cont}$ for every non-compact $σ$-compact locally compact abelian group. These results answer Questions 6.1--6.3 and 6.5--6.7, and the locally compact case of Question 6.4, posed by L.~Außenhofer and D.~Dikranjan in \\cite{AD20}. The embedding also applies to non-compact complete metrizable locally quasi-convex groups. A non-compact precompact nuclear group with a unique compatible topology shows that non-compactness alone does not suffice in the nuclear setting. Finally, the compatible poset of $\\R^{\\N}$ is not order-isomorphic to that of any discrete abelian group.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12116",
  "author_total": 1,
  "authors": [
    {
      "name": "ZhouXiang Huang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4tar2kjc2sabn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12116",
  "fulltext_url": null,
  "id": "4tar2kjc2sabn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12116",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.GN"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7221170189",
  "pdf_url": "https://arxiv.org/pdf/2610.12116",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4tar2kjc2sabn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.12116",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12116"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The size of the poset of compatible locally quasi-convex topologies on locally compact abelian groups",
  "title_text": "The size of the poset of compatible locally quasi-convex topologies on locally compact abelian groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/4tar2kjc2sabn",
  "venue": "arXiv",
  "versions": []
}